Ranicki, Andrew Algebraic and geometric splittings of the K- and L-groups of polynomial extensions. (English) Zbl 0615.57017 Transformation groups, Proc. Symp., Poznań/Pol. 1985, Lect. Notes Math. 1217, 321-363 (1986). [For the entire collection see Zbl 0594.00014.] For L-groups of polynomial extensions one has splittings \(L^ s_*(\pi \times {\mathbb{Z}})=L^ s_*(\pi)\oplus L^ h_{*-1}(\pi)\) and \(L^ h_*(\pi \times {\mathbb{Z}})=L^ h_*(\pi)\oplus L^ p_{*-1}(\pi)\). Such splittings were obtained geometrically by J. L. Shaneson [Ann. Math., II. Ser. 90, 296-334 (1969; Zbl 0182.573)] and E. K. Pedersen and the author [Topology 19, 239-254 (1980; Zbl 0477.57020)] and algebraically by S. P. Novikov [Izv. Akad. Nauk SSSR, Ser. Mat. 34, 253-288 (1970; Zbl 0193.519)] and the author [Proc. Lond. Math. Soc., III. Ser. 27, 101-158 (1973; Zbl 0269.18009)]. In the present paper the author shows that these splittings in general do not coincide, and he expresses the difference, which is always 2-torsion, in terms of algebra. A similar result holds for geometric and algebraic splittings of the Whitehead group of a polynomial extension. Reviewer: M.Kolster Cited in 5 Documents MSC: 57R67 Surgery obstructions, Wall groups 57Q10 Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc. 18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects) Keywords:L-groups of polynomial extensions; geometric and algebraic splittings; Whitehead group of a polynomial extension Citations:Zbl 0216.450; Zbl 0594.00014; Zbl 0182.573; Zbl 0477.57020; Zbl 0193.519; Zbl 0269.18009 × Cite Format Result Cite Review PDF